Boston school buses — chronically late last year, prompting a barrage of criticism from students, parents, and teachers ­— are increasingly arriving on time this fall.

Since the third week of school, at least 91 percent of buses have arrived each week before the morning bell, according to School Department data. For the most part, the on-time performance rates have steadily increased each week, hitting a high of 96.1 percent Thanksgiving week.

That is in sharp contrast to last fall, when on-time rates were below 80 percent for much of that time, causing the School Committee to declare a crisis as buses ran as much as an hour late.

“We keep getting better,” said Carl Allen, the school department’s transportation director. “The late buses we do see are because of traffic or the occasional accident, road construction, or an issue with a student.”

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Superintendent Carol R. Johnson, who undertook an overhaul of the busing system last year, enacted a number of changes to improve the buses, most notably naming a new transportation director, drawing up routes earlier, and cross-referencing computer-generated travel times with drivers’ knowledge of the streets and problem intersections.

Officials point to the improved performance as evidence that they have finally figured out how to devise routes that better reflect the time it truly takes to traverse the city’s traffic-clogged streets during rush hour.

Steve Kirschbaum, a driver and chairman of the grievance committee for the school bus drivers union, said some routing problems persist — causing the union to file two class-action grievances and many individual grievances on specific routes — and the School Department has been routinely changing routes.

“If you are doing routing that makes sense you won’t have runs with 18 stops,” Kirschbaum said. “There are regular routing and across-the-board changes that are being made to the routes for cost efficiencies at the expense of safe, on-time transportation of students.”

But Dumond Louis, president of the bus drivers union, downplayed the issues.

“I think we are doing pretty good this year,” Louis said. “There will be issues from time to time.”

Allen said any changes to routes are done to improve service for students and sometimes at a parent’s request who wants a different drop-off location for their child after school. He noted the School Department added about 500 routes this year in an effort to improve arrival times.

The department operates a massive transportation system, shuttling more than 33,000 students each day to 228 public and private schools. To execute this daily journey, the School Department dispatches more than 700 buses, which cover more than 3,600 routes.

The operation, largely overseen by a bus management company, costs the district about $80 million a year.

Tardy buses first became a major issue in fall 2010. The problem was primarily because of routes generated through a recently implemented Web-based software program that grossly undercalculated the travel times.

The School Department eventually fixed those routes, but then chaos of even larger proportions broke out last fall as school officials confronted what they called a “perfect storm.” In the months leading up to that school year, as it was working through problems from the software program, the department consolidated more than 1,500 routes and closed and merged several schools, and new charter schools were opened.

Making matters worse, school officials shortened the time frame to create, test, and modify the bus routes. School and city officials also blamed some bus drivers for showing up late to work, an accusation contested by the drivers union.

By October, a frustrated Thomas M. Menino declared the busing system “broken” and appointed a top aide to intervene, setting in motion the efforts to fix the problem.

This school year started off on a strong note, with 87 percent of buses arriving on time during the first full week of school, compared with 63 percent for the same week the previous year.

To boost the arrival time, Allen said the department focuses intensely on chronically late buses. The reasons buses are tardy tend to vary from one week to another. Many times a new issue emerges, such as a road project, a substitute filling in for a sick driver, or the addition of a newly registered student to a route.

Late buses also tend to travel some of the longest distances. Boston’s student-assignment system allows students from East Boston to attend schools in Brighton, for instance.

Most of the late buses arrive no more than 10 minutes after the morning bell. Few, if any, buses in a given week are more than a half hour late, a reversal from last fall.

With more students arriving to school on time, academic achievement could improve, said the Rev. Gregory Groover, chairman of the Boston School Committee.

“Students are getting an opportunity to eat breakfast, they are more energized, and there is less anxiousness about being late,” said Groover, also noting that teachers can start instruction right away because the buses are not late. “All that plays a role . . . in improving children’s academic performance.”

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